HMMT 2020

Let $P(x)=x^{3}+x^{2}-r^{2} x-2020$ be a polynomial with roots $r, s, t$. What is $P(1)$ ?

Find the unique pair of positive integers $(a, b)$ with $a<b$ for which

$$\frac{2020-a}{a} \cdot \frac{2020-b}{b}=2$$

Let $a=256$. Find the unique real number $x>a^{2}$ such that

$$\log _{a} \log _{a} \log _{a} x=\log _{a^{2}} \log _{a^{2}} \log _{a^{2}} x$$

For positive integers $n$ and $k$, let $\mho(n, k)$ be the number of distinct prime divisors of $n$ that are at least $k$. For example, $\mho(90,3)=2$, since the only prime factors of 90 that are at least 3 are 3 and 5 . Find the closest integer to

$$\sum_{n=1}^{\infty} \sum_{k=1}^{\infty} \frac{\mho(n, k)}{3^{n+k-7}}$$

A positive integer $N$ is piquant if there exists a positive integer $m$ such that if $n_{i}$ denotes the number of digits in $m^{i}$ (in base 10), then $n_{1}+n_{2}+\cdots+n_{10}=N$. Let $p_{M}$ denote the fraction of the first $M$ positive integers that are piquant. Find $\lim _{M \rightarrow \infty} p_{M}$.

A polynomial $P(x)$ is a base-n polynomial if it is of the form $a_{d} x^{d}+a_{d-1} x^{d-1}+\cdots+a_{1} x+a_{0}$, where each $a_{i}$ is an integer between 0 and $n-1$ inclusive and $a_{d}>0$. Find the largest positive integer $n$ such that for any real number $c$, there exists at most one base- $n$ polynomial $P(x)$ for which $P(\sqrt{2}+\sqrt{3})=c$.

Find the sum of all positive integers $n$ for which

$$\frac{15 \cdot n!^{2}+1}{2 n-3}$$

is an integer.

Let $P(x)$ be the unique polynomial of degree at most 2020 satisfying $P\left(k^{2}\right)=k$ for $k=0,1,2, \ldots, 2020$. Compute $P\left(2021^{2}\right)$.

Let $P(x)=x^{2020}+x+2$, which has 2020 distinct roots. Let $Q(x)$ be the monic polynomial of degree $\binom{2020}{2}$ whose roots are the pairwise products of the roots of $P(x)$. Let $\alpha$ satisfy $P(\alpha)=4$. Compute the sum of all possible values of $Q\left(\alpha^{2}\right)^{2}$.

We define $\mathbb{F}_{101}[x]$ as the set of all polynomials in $x$ with coefficients in $\mathbb{F}_{101}$ (the integers modulo 101 with usual addition and subtraction), so that two polynomials are equal if and only if the coefficients of $x^{k}$ are equal in $\mathbb{F}_{101}$ for each nonnegative integer $k$. For example, $(x+3)(100 x+5)=100 x^{2}+2 x+15$ in $\mathbb{F}_{101}[x]$ because the corresponding coefficients are equal modulo 101. We say that $f(x) \in \mathbb{F}_{101}[x]$ is lucky if it has degree at most 1000 and there exist $g(x), h(x) \in \mathbb{F}_{101}[x]$ such that

$$f(x)=g(x)\left(x^{1001}-1\right)+h(x)^{101}-h(x)$$

in $\mathbb{F}_{101}[x]$. Find the number of lucky polynomials.

Let $D I A L, F O R$, and $F R I E N D$ be regular polygons in the plane. If $I D=1$, find the product of all possible areas of $O L A$.

Let $A B C$ be a triangle with $A B=5, A C=8$, and $\angle B A C=60^{\circ}$. Let $U V W X Y Z$ be a regular hexagon that is inscribed inside $A B C$ such that $U$ and $V$ lie on side $B A, W$ and $X$ lie on side $A C$, and $Z$ lies on side $C B$. What is the side length of hexagon $U V W X Y Z$ ?

Consider the L-shaped tromino below with 3 attached unit squares. It is cut into exactly two pieces of equal area by a line segment whose endpoints lie on the perimeter of the tromino. What is the longest possible length of the line segment?

Let $A B C D$ be a rectangle and $E$ be a point on segment $A D$. We are given that quadrilateral $B C D E$ has an inscribed circle $\omega_{1}$ that is tangent to $B E$ at $T$. If the incircle $\omega_{2}$ of $A B E$ is also tangent to $B E$ at $T$, then find the ratio of the radius of $\omega_{1}$ to the radius of $\omega_{2}$.

Let $A B C D E F$ be a regular hexagon with side length 2. A circle with radius 3 and center at $A$ is drawn. Find the area inside quadrilateral $B C D E$ but outside the circle.

Let $A B C$ be a triangle with $A B=5, B C=6, C A=7$. Let $D$ be a point on ray $A B$ beyond $B$ such that $B D=7, E$ be a point on ray $B C$ beyond $C$ such that $C E=5$, and $F$ be a point on ray $C A$ beyond $A$ such that $A F=6$. Compute the area of the circumcircle of $D E F$.

Let $\Gamma$ be a circle, and $\omega_{1}$ and $\omega_{2}$ be two non-intersecting circles inside $\Gamma$ that are internally tangent to $\Gamma$ at $X_{1}$ and $X_{2}$, respectively. Let one of the common internal tangents of $\omega_{1}$ and $\omega_{2}$ touch $\omega_{1}$ and $\omega_{2}$ at $T_{1}$ and $T_{2}$, respectively, while intersecting $\Gamma$ at two points $A$ and $B$. Given that $2 X_{1} T_{1}=X_{2} T_{2}$ and that $\omega_{1}, \omega_{2}$, and $\Gamma$ have radii 2, 3, and 12 , respectively, compute the length of $A B$.

Let $A B C$ be an acute triangle with circumcircle $\Gamma$. Let the internal angle bisector of $\angle B A C$ intersect $B C$ and $\Gamma$ at $E$ and $N$, respectively. Let $A^{\prime}$ be the antipode of $A$ on $\Gamma$ and let $V$ be the point where $A A^{\prime}$ intersects $B C$. Given that $E V=6, V A^{\prime}=7$, and $A^{\prime} N=9$, compute the radius of $\Gamma$.

Circles $\omega_{a}, \omega_{b}, \omega_{c}$ have centers $A, B, C$, respectively and are pairwise externally tangent at points $D, E, F$ (with $D \in B C, E \in C A, F \in A B$ ). Lines $B E$ and $C F$ meet at $T$. Given that $\omega_{a}$ has radius 341 , there exists a line $\ell$ tangent to all three circles, and there exists a circle of radius 49 tangent to all three circles, compute the distance from $T$ to $\ell$.

Let $\Gamma$ be a circle of radius 1 centered at $O$. A circle $\Omega$ is said to be friendly if there exist distinct circles $\omega_{1}, \omega_{2}, \ldots, \omega_{2020}$, such that for all $1 \leq i \leq 2020, \omega_{i}$ is tangent to $\Gamma, \Omega$, and $\omega_{i+1}$. (Here, $\omega_{2021}=\omega_{1}$.) For each point $P$ in the plane, let $f(P)$ denote the sum of the areas of all friendly circles centered at $P$. If $A$ and $B$ are points such that $O A=\frac{1}{2}$ and $O B=\frac{1}{3}$, determine $f(A)-f(B)$.

How many ways can the vertices of a cube be colored red or blue so that the color of each vertex is the color of the majority of the three vertices adjacent to it?

How many positive integers at most 420 leave different remainders when divided by each of 5,6 , and 7 ?

Each unit square of a $4 \times 4$ square grid is colored either red, green, or blue. Over all possible colorings of the grid, what is the maximum possible number of L-trominos that contain exactly one square of each color? (L-trominos are made up of three unit squares sharing a corner, as shown below.)

$$\square \square \square \square \square \square \square$$

Given an $8 \times 8$ checkerboard with alternating white and black squares, how many ways are there to choose four black squares and four white squares so that no two of the eight chosen squares are in the same row or column?

Let $S$ be a set of intervals defined recursively as follows:

• Initially, $[1,1000]$ is the only interval in $S$.
• If $l \neq r$ and $[l, r] \in S$, then both $\left[l,\left\lfloor\frac{l+r}{2}\right\rfloor\right],\left[\left\lfloor\frac{l+r}{2}\right\rfloor+1, r\right] \in S$. (Note that $S$ can contain intervals such as [1, 1], which contain a single integer.) An integer $i$ is chosen uniformly at random from the range [1,1000]. What is the expected number of intervals in $S$ which contain $i$ ?

Alice writes 1001 letters on a blackboard, each one chosen independently and uniformly at random from the set $S=\{a, b, c\}$. A move consists of erasing two distinct letters from the board and replacing them with the third letter in $S$. What is the probability that Alice can perform a sequence of moves which results in one letter remaining on the blackboard?

Anne-Marie has a deck of 16 cards, each with a distinct positive factor of 2002 written on it. She shuffles the deck and begins to draw cards from the deck without replacement. She stops when there exists a nonempty subset of the cards in her hand whose numbers multiply to a perfect square. What is the expected number of cards in her hand when she stops?

Let $\Gamma_{1}$ and $\Gamma_{2}$ be concentric circles with radii 1 and 2 , respectively. Four points are chosen on the circumference of $\Gamma_{2}$ independently and uniformly at random, and are then connected to form a convex quadrilateral. What is the probability that the perimeter of this quadrilateral intersects $\Gamma_{1}$ ?

Farmer James wishes to cover a circle with circumference $10 \pi$ with six different types of colored arcs. Each type of arc has radius 5 , has length either $\pi$ or $2 \pi$, and is colored either red, green, or blue. He has an unlimited number of each of the six arc types. He wishes to completely cover his circle without overlap, subject to the following conditions:

• Any two adjacent arcs are of different colors.
• Any three adjacent arcs where the middle arc has length $\pi$ are of three different colors.

Find the number of distinct ways Farmer James can cover his circle. Here, two coverings are equivalent if and only if they are rotations of one another. In particular, two colorings are considered distinct if they are reflections of one another, but not rotations of one another.

Max repeatedly throws a fair coin in a hurricane. For each throw, there is a $4 \%$ chance that the coin gets blown away. He records the number of heads $H$ and the number of tails $T$ before the coin is lost. (If the coin is blown away on a toss, no result is recorded for that toss.) What is the expected value of $|H-T|$ ?

Let $n$ be a positive integer. Define a sequence by $a_{0}=1, a_{2 i+1}=a_{i}$, and $a_{2 i+2}=a_{i}+a_{i+1}$ for each $i \geq 0$. Determine, with proof, the value of $a_{0}+a_{1}+a_{2}+\cdots+a_{2^{n}-1}$.

Let $n$ be a fixed positive integer. An $n$-staircase is a polyomino with $\frac{n(n+1)}{2}$ cells arranged in the shape of a staircase, with arbitrary size. Here are two examples of 5 -staircases:

Prove that an $n$-staircase can be dissected into strictly smaller $n$-staircases.

Let $A B C$ be a triangle inscribed in a circle $\omega$ and $\ell$ be the tangent to $\omega$ at $A$. The line through $B$ parallel to $A C$ meets $\ell$ at $P$, and the line through $C$ parallel to $A B$ meets $\ell$ at $Q$. The circumcircles of $A B P$ and $A C Q$ meet at $S \neq A$. Show that $A S$ bisects $B C$.

Alan draws a convex 2020-gon $\mathcal{A}=A_{1} A_{2} \cdots A_{2020}$ with vertices in clockwise order and chooses 2020 angles $\theta_{1}, \theta_{2}, \ldots, \theta_{2020} \in(0, \pi)$ in radians with sum $1010 \pi$. He then constructs isosceles triangles $\triangle A_{i} B_{i} A_{i+1}$ on the exterior of $\mathcal{A}$ with $B_{i} A_{i}=B_{i} A_{i+1}$ and $\angle A_{i} B_{i} A_{i+1}=\theta_{i}$. (Here, $A_{2021}=A_{1}$.) Finally, he erases $\mathcal{A}$ and the point $B_{1}$. He then tells Jason the angles $\theta_{1}, \theta_{2}, \ldots, \theta_{2020}$ he chose. Show that Jason can determine where $B_{1}$ was from the remaining 2019 points, i.e. show that $B_{1}$ is uniquely determined by the information Jason has.

Let $a_{0}, b_{0}, c_{0}, a, b, c$ be integers such that $\operatorname{gcd}\left(a_{0}, b_{0}, c_{0}\right)=\operatorname{gcd}(a, b, c)=1$. Prove that there exists a positive integer $n$ and integers $a_{1}, a_{2}, \ldots, a_{n}=a, b_{1}, b_{2}, \ldots, b_{n}=b, c_{1}, c_{2}, \ldots, c_{n}=c$ such that for all $1 \leq i \leq n, a_{i-1} a_{i}+b_{i-1} b_{i}+c_{i-1} c_{i}=1$.

Let $n>1$ be a positive integer and $S$ be a collection of $\frac{1}{2}\binom{2 n}{n}$ distinct $n$-element subsets of $\{1,2, \ldots, 2 n\}$. Show that there exists $A, B \in S$ such that $|A \cap B| \leq 1$.

Positive real numbers $x$ and $y$ satisfy

$$||\cdots|||x|-y|-x| \cdots-y|-x|=||\cdots|||y|-x|-y| \cdots-x|-y|$$

where there are 2019 absolute value signs $|\cdot|$ on each side. Determine, with proof, all possible values of $\frac{x}{y}$.

Let $A B C$ be a scalene triangle with angle bisectors $A D, B E$, and $C F$ so that $D, E$, and $F$ lie on segments $B C, C A$, and $A B$ respectively. Let $M$ and $N$ be the midpoints of $B C$ and $E F$ respectively. Prove that line $A N$ and the line through $M$ parallel to $A D$ intersect on the circumcircle of $A B C$ if and only if $D E=D F$.

Let $p>5$ be a prime number. Show that there exists a prime number $q<p$ and a positive integer $n$ such that $p$ divides $n^{2}-q$.

Let $n$ be a fixed positive integer, and choose $n$ positive integers $a_{1}, \ldots, a_{n}$. Given a permutation $\pi$ on the first $n$ positive integers, let $S_{\pi}=\left\{i \left\lvert \frac{a_{i}}{\pi(i)}\right.\right.$ is an integer $\}$. Let $N$ denote the number of distinct sets $S_{\pi}$ as $\pi$ ranges over all such permutations. Determine, in terms of $n$, the maximum value of $N$ over all possible values of $a_{1}, \ldots, a_{n}$.

Submit an integer $x$ as your answer to this problem. The number of points you receive will be $\max (0,8-|8 x-100|)$. (Non-integer answers will be given 0 points.)

Let $A B C$ be a triangle and $\omega$ be its circumcircle. The point $M$ is the midpoint of arc $B C$ not containing $A$ on $\omega$ and $D$ is chosen so that $D M$ is tangent to $\omega$ and is on the same side of $A M$ as $C$. It is given that $A M=A C$ and $\angle D M C=38^{\circ}$. Find the measure of angle $\angle A C B$.

Let $A B C$ be a triangle and $D, E$, and $F$ be the midpoints of sides $B C, C A$, and $A B$ respectively. What is the maximum number of circles which pass through at least 3 of these 6 points?

Compute the value of $\sqrt{105^{3}-104^{3}}$, given that it is a positive integer.

Alice, Bob, and Charlie roll a 4,5 , and 6 -sided die, respectively. What is the probability that a number comes up exactly twice out of the three rolls?

Two sides of a regular $n$-gon are extended to meet at a $28^{\circ}$ angle. What is the smallest possible value for $n$ ?

Ana and Banana are rolling a standard six-sided die. Ana rolls the die twice, obtaining $a_{1}$ and $a_{2}$, then Banana rolls the die twice, obtaining $b_{1}$ and $b_{2}$. After Ana's two rolls but before Banana's two rolls, they compute the probability $p$ that $a_{1} b_{1}+a_{2} b_{2}$ will be a multiple of 6 . What is the probability that $p=\frac{1}{6}$ ?

Tessa picks three real numbers $x, y, z$ and computes the values of the eight expressions of the form $\pm x \pm y \pm z$. She notices that the eight values are all distinct, so she writes the expressions down in increasing order. For example, if $x=2, y=3, z=4$, then the order she writes them down is

$$-x-y-z,+x-y-z,-x+y-z,-x-y+z,+x+y-z,+x-y+z,-x+y+z,+x+y+z$$

How many possible orders are there?

Let $P(x)$ be the monic polynomial with rational coefficients of minimal degree such that $\frac{1}{\sqrt{2}}$, $\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{4}}, \ldots, \frac{1}{\sqrt{1000}}$ are roots of $P$. What is the sum of the coefficients of $P$ ?

Jarris is a weighted tetrahedral die with faces $F_{1}, F_{2}, F_{3}, F_{4}$. He tosses himself onto a table, so that the probability he lands on a given face is proportional to the area of that face (i.e. the probability he lands on face $F_{i}$ is $\frac{\left[F_{i}\right]}{\left[F_{1}\right]+\left[F_{2}\right]+\left[F_{3}\right]+\left[F_{4}\right]}$ where $[K]$ is the area of $\left.K\right)$. Let $k$ be the maximum distance any part of Jarris is from the table after he rolls himself. Given that Jarris has an inscribed sphere of radius 3 and circumscribed sphere of radius 10, find the minimum possible value of the expected value of $k$.

Find the number of ordered pairs of positive integers $(x, y)$ with $x, y \leq 2020$ such that $3 x^{2}+10 x y+$ $3 y^{2}$ is the power of some prime.

An $11 \times 11$ grid is labeled with consecutive rows $0,1,2, \ldots, 10$ and columns $0,1,2, \ldots, 10$ so that it is filled with integers from 1 to $2^{10}$, inclusive, and the sum of all of the numbers in row $n$ and in column $n$ are both divisible by $2^{n}$. Find the number of possible distinct grids.

Let $\triangle A B C$ be a triangle with $A B=7, B C=1$, and $C A=4 \sqrt{3}$. The angle trisectors of $C$ intersect $\overline{A B}$ at $D$ and $E$, and lines $\overline{A C}$ and $\overline{B C}$ intersect the circumcircle of $\triangle C D E$ again at $X$ and $Y$, respectively. Find the length of $X Y$.

Let $\varphi(n)$ denote the number of positive integers less than or equal to $n$ which are relatively prime to $n$. Let $S$ be the set of positive integers $n$ such that $\frac{2 n}{\varphi(n)}$ is an integer. Compute the sum

$$\sum_{n \in S} \frac{1}{n}$$

You have six blocks in a row, labeled 1 through 6 , each with weight 1 . Call two blocks $x \leq y$ connected when, for all $x \leq z \leq y$, block $z$ has not been removed. While there is still at least one block remaining, you choose a remaining block uniformly at random and remove it. The cost of this operation is the sum of the weights of the blocks that are connected to the block being removed, including itself. Compute the expected total cost of removing all the blocks.

Determine all triplets of real numbers $(x, y, z)$ satisfying the system of equations

$$\begin{aligned} x^{2} y+y^{2} z & =1040 \\ x^{2} z+z^{2} y & =260 \\ (x-y)(y-z)(z-x) & =-540 \end{aligned}$$

Let $A B C$ be a triangle with incircle tangent to the perpendicular bisector of $B C$. If $B C=A E=$ 20, where $E$ is the point where the $A$-excircle touches $B C$, then compute the area of $\triangle A B C$.

A vertex-induced subgraph is a subset of the vertices of a graph together with any edges whose endpoints are both in this subset. An undirected graph contains 10 nodes and $m$ edges, with no loops or multiple edges. What is the minimum possible value of $m$ such that this graph must contain a nonempty vertex-induced subgraph where all vertices have degree at least 5 ?

The Fibonacci numbers are defined by $F_{0}=0, F_{1}=1$, and $F_{n}=F_{n-1}+F_{n-2}$ for $n \geq 2$. There exist unique positive integers $n_{1}, n_{2}, n_{3}, n_{4}, n_{5}, n_{6}$ such that

$$\sum_{i_{1}=0}^{100} \sum_{i_{2}=0}^{100} \sum_{i_{3}=0}^{100} \sum_{i_{4}=0}^{100} \sum_{i_{5}=0}^{100} F_{i_{1}+i_{2}+i_{3}+i_{4}+i_{5}}=F_{n_{1}}-5 F_{n_{2}}+10 F_{n_{3}}-10 F_{n_{4}}+5 F_{n_{5}}-F_{n_{6}}$$

Find $n_{1}+n_{2}+n_{3}+n_{4}+n_{5}+n_{6}$.

There exist several solutions to the equation

$$1+\frac{\sin x}{\sin 4 x}=\frac{\sin 3 x}{\sin 2 x}$$

where $x$ is expressed in degrees and $0^{\circ}<x<180^{\circ}$. Find the sum of all such solutions.

We call a positive integer $t$ good if there is a sequence $a_{0}, a_{1}, \ldots$ of positive integers satisfying $a_{0}=15, a_{1}=t$, and

$$a_{n-1} a_{n+1}=\left(a_{n}-1\right)\left(a_{n}+1\right)$$

for all positive integers $n$. Find the sum of all good numbers.

Let $A$ be a set of integers such that for each integer $m$, there exists an integer $a \in A$ and positive integer $n$ such that $a^{n} \equiv m(\bmod 100)$. What is the smallest possible value of $|A|$ ?

A function $f: A \rightarrow A$ is called idempotent if $f(f(x))=f(x)$ for all $x \in A$. Let $I_{n}$ be the number of idempotent functions from $\{1,2, \ldots, n\}$ to itself. Compute

$$\sum_{n=1}^{\infty} \frac{I_{n}}{n!}$$

In $\triangle A B C, \omega$ is the circumcircle, $I$ is the incenter and $I_{A}$ is the $A$-excenter. Let $M$ be the midpoint of arc $\widehat{B A C}$ on $\omega$, and suppose that $X, Y$ are the projections of $I$ onto $M I_{A}$ and $I_{A}$ onto $M I$, respectively. If $\triangle X Y I_{A}$ is an equilateral triangle with side length 1 , compute the area of $\triangle A B C$.

Let $S$ be the set of $3^{4}$ points in four-dimensional space where each coordinate is in $\{-1,0,1\}$. Let $N$ be the number of sequences of points $P_{1}, P_{2}, \ldots, P_{2020}$ in $S$ such that $P_{i} P_{i+1}=2$ for all $1 \leq i \leq 2020$ and $P_{1}=(0,0,0,0)$. (Here $P_{2021}=P_{1}$.) Find the largest integer $n$ such that $2^{n}$ divides $N$.

Let $A B C D$ be a cyclic quadrilateral, and let segments $A C$ and $B D$ intersect at $E$. Let $W$ and $Y$ be the feet of the altitudes from $E$ to sides $D A$ and $B C$, respectively, and let $X$ and $Z$ be the midpoints of sides $A B$ and $C D$, respectively. Given that the area of $A E D$ is 9 , the area of $B E C$ is 25 , and $\angle E B C-\angle E C B=30^{\circ}$, then compute the area of $W X Y Z$.

Let $\left\{a_{i}\right\}_{i \geq 0}$ be a sequence of real numbers defined by

$$a_{n+1}=a_{n}^{2}-\frac{1}{2^{2020 \cdot 2^{n}-1}}$$

for $n \geq 0$. Determine the largest value for $a_{0}$ such that $\left\{a_{i}\right\}_{i \geq 0}$ is bounded.

Let $\triangle A B C$ be a triangle inscribed in a unit circle with center $O$. Let $I$ be the incenter of $\triangle A B C$, and let $D$ be the intersection of $B C$ and the angle bisector of $\angle B A C$. Suppose that the circumcircle of $\triangle A D O$ intersects $B C$ again at a point $E$ such that $E$ lies on $I O$. If $\cos A=\frac{12}{13}$, find the area of $\triangle A B C$.

Let $A B C D$ be a tetrahedron such that its circumscribed sphere of radius $R$ and its inscribed sphere of radius $r$ are concentric. Given that $A B=A C=1 \leq B C$ and $R=4 r$, find $B C^{2}$.

Let $S=\{(x, y) \mid x>0, y>0, x+y<200$, and $x, y \in \mathbb{Z}\}$. Find the number of parabolas $\mathcal{P}$ with vertex $V$ that satisfy the following conditions:

• $\mathcal{P}$ goes through both $(100,100)$ and at least one point in $S$,
• $V$ has integer coordinates, and
• $\mathcal{P}$ is tangent to the line $x+y=0$ at $V$.

Anastasia is taking a walk in the plane, starting from $(1,0)$. Each second, if she is at $(x, y)$, she moves to one of the points $(x-1, y),(x+1, y),(x, y-1)$, and $(x, y+1)$, each with $\frac{1}{4}$ probability. She stops as soon as she hits a point of the form $(k, k)$. What is the probability that $k$ is divisible by 3 when she stops?

Find the smallest real constant $\alpha$ such that for all positive integers $n$ and real numbers $0=y_{0}<$ $y_{1}<\cdots<y_{n}$, the following inequality holds:

$$\alpha \sum_{k=1}^{n} \frac{(k+1)^{3 / 2}}{\sqrt{y_{k}^{2}-y_{k-1}^{2}}} \geq \sum_{k=1}^{n} \frac{k^{2}+3 k+3}{y_{k}} \text {. }$$

Estimate

$$N=\prod_{n=1}^{\infty} n^{n^{-1.25}}$$

An estimate of $E>0$ will receive $\lfloor 22 \min (N / E, E / N)\rfloor$ points.

For odd primes $p$, let $f(p)$ denote the smallest positive integer $a$ for which there does not exist an integer $n$ satisfying $p \mid n^{2}-a$. Estimate $N$, the sum of $f(p)^{2}$ over the first $10^{5}$ odd primes $p$. An estimate of $E>0$ will receive $\left\lfloor 22 \min (N / E, E / N)^{3}\right\rfloor$ points.

A collection $\mathcal{S}$ of 10000 points is formed by picking each point uniformly at random inside a circle of radius 1 . Let $N$ be the expected number of points of $\mathcal{S}$ which are vertices of the convex hull of the $\mathcal{S}$. (The convex hull is the smallest convex polygon containing every point of $\mathcal{S}$.) Estimate $N$. An estimate of $E>0$ will earn $\max (\lfloor 22-|E-N|\rfloor, 0)$ points.

A snake of length $k$ is an animal which occupies an ordered $k$-tuple $\left(s_{1}, \ldots, s_{k}\right)$ of cells in a $n \times n$ grid of square unit cells. These cells must be pairwise distinct, and $s_{i}$ and $s_{i+1}$ must share a side for $i=1, \ldots, k-1$. If the snake is currently occupying $\left(s_{1}, \ldots, s_{k}\right)$ and $s$ is an unoccupied cell sharing a side with $s_{1}$, the snake can move to occupy $\left(s, s_{1}, \ldots, s_{k-1}\right)$ instead. Initially, a snake of length 4 is in the grid $\{1,2, \ldots, 30\}^{2}$ occupying the positions $(1,1),(1,2),(1,3),(1,4)$ with $(1,1)$ as its head. The snake repeatedly makes a move uniformly at random among moves it can legally make. Estimate $N$, the expected number of moves the snake makes before it has no legal moves remaining.

An estimate of $E>0$ will earn $\left\lfloor 22 \min (N / E, E / N)^{4}\right\rfloor$ points.